where A and B are fixed polynomials with degA ≤ 2, and degB = 1. In 1939, Shohat extended these ideas introducing a new class of orthogonal polynomials. In fact, he studied orthogonal polynomials associated with forms satisfying the last equation, with no restrictions in the degrees of the polynomials A, and B. Obviously, orthogonal polynomials defined as above, generalize in a natural way the ...

Hikita polynomials are the combinatorial side of the rational shuffle theorem. Building upon a recent formula for (m, 3)-Catalan polynomials, we prove a formula for (m, 3)-Hikita polynomials in terms of Catalan polynomials. This formula shows a surprising relation among coefficients of Hikita polynomials and implies deeper recursive relations and proves the q, t-symmetry of (m, 3)-Hikita polyno...

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the Eulerian polynomials and the shifted Eulerian polynomials are moment sequences for a simple family of orthogonal polynomials. The coefficient arrays of these families of orthogonal polynomials are shown to be exponential Riordan arrays. Using the theory of orthogonal polynomials we are then able t...

We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irre-ducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree n polynomials of Hamming distance one from a randomly chosen set of 2 n /n ...

We study integer coefficient polynomials of fixed degree and maximum height H that are irreducible by the Dumas criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials as H → ∞. We also show that, for a fixed degree, the density of Dumas polynomials in the set of all irreducible integer coefficient polynomials is strictly less than 1.

in this paper, we propose and analyze an efficient matrix methodbased on bell polynomials for numerically solving nonlinear fredholm- volterraintegral equations. for this aim, first we calculate operational matrix of integration and product based on bell polynomials. by using these matrices, nonlinearfredholm-volterra integral equations reduce to the system of nonlinear algebraicequations which...

In this paper, we consider the problem of representing any polynomial in terms degenerate Bernoulli polynomials and more generally higher-order polynomials. We derive explicit formulas with help umbral calculus illustrate our results some examples.